What Do the Derivatives and Graphs Reveal About y=(2x+1)/\sqrt{x^2+1}?

[Hurricane3]

Homework Statement

y=(2x+1)/\sqrt{x^2+1}

Find where are the asymptotes, where is it increasing increasing/decreasing, ect...

Homework Equations

The Attempt at a Solution

when I took the first derivative (im trying to find where it increases/decrease), I got
dy/dx = \frac{2x^2-x+2}{(x^2+1)\sqrt{x^2+1}}

but there isn't any roots for this function... so what does that mean?

 

[CocaCola]

No vertical asymptotes as x^2+1 can never be 0.

No horizontal asy as (x^2)^1/2 is one, and 2x is one.

Slant asymptote is synthetic division (x^2+1)^(1/2) |2x+1

If there are no 0's then that means that the equation always increases or decreases.

 

[HallsofIvy]

Grammer police: Better would be "either always increases or always decreases".


Any graph that does not have a horizontal line segment "always increases or decreases"!